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Properties

Label 4-15e4-1.1-c0e2-0-0
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $0.0126089$
Root an. cond. $0.335096$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16-s − 4·31-s + 4·61-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 16-s − 4·31-s + 4·61-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.0126089\)
Root analytic conductor: \(0.335096\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5053588586\)
\(L(\frac12)\) \(\approx\) \(0.5053588586\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$ \( ( 1 + T )^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$ \( ( 1 - T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.69803762574422672434967233354, −12.37548083065245157988382609152, −11.56606673163933529207753982522, −11.31462212756581432986790872170, −10.90829519439838618163105865185, −10.39128633790638138643671563795, −9.662959581269237668744357861175, −9.419581453182552381230800138331, −8.679914944651945921251544356863, −8.558663126224290290571978918038, −7.53069093482151041493926454554, −7.30990867419776323575653941912, −6.73232390186078945006518717104, −6.09326950367494740805627823014, −5.26331522599767078774420425881, −5.13878211294140034503208333172, −3.88505578156172457214518408787, −3.79449030796393505024174188555, −2.57334003132524961343902120842, −1.82341514415177521088400579334, 1.82341514415177521088400579334, 2.57334003132524961343902120842, 3.79449030796393505024174188555, 3.88505578156172457214518408787, 5.13878211294140034503208333172, 5.26331522599767078774420425881, 6.09326950367494740805627823014, 6.73232390186078945006518717104, 7.30990867419776323575653941912, 7.53069093482151041493926454554, 8.558663126224290290571978918038, 8.679914944651945921251544356863, 9.419581453182552381230800138331, 9.662959581269237668744357861175, 10.39128633790638138643671563795, 10.90829519439838618163105865185, 11.31462212756581432986790872170, 11.56606673163933529207753982522, 12.37548083065245157988382609152, 12.69803762574422672434967233354

Graph of the $Z$-function along the critical line